Our lovely little Roborovski dwarf hamsters, Yao and Shun, were named after two of the sage kings of Chinese antiquity. We’re into Asian history, and the little white eyebrows made them look like little old kung-fu masters.
Sophie, at the age of five, named the new Scottish Terrier puppy “Mackie”, with no prodding from me or her mother. I am still impressed by how she did that.
We adopted an orange calico (is that the right term?) cat. We have oak floors in the house, and when we brought the cat home, the first thing then 8-year-old SqueegeeJr noticed was how similar the cat’s coat was to our flooring, and he proposed naming the cat “Wood”. My clever wife immediately riposted that we could name the cat with the Spanish name for wood, “madera”. SqueegeeJr found that acceptable, and “La Madera”, or “Dera” is her name to this day.
Merlyn came up to me one day and decided he was going to stick around. I wanted a name suggesting magic (he was going to be the cat of a witch, after all), so my neighbor suggested the name. Came to fit him even more than I would have guessed – he had some scary health issues a couple years ago which first nearly killed him, then turned into something fairly benign as if by magic. He also regularly takes it upon himself to be there with me during trancework; don’t ask me how he knows, he just does.
Púca I got as a 12-week-old kitten. I wanted an Irish/Gaelic related name. So I did some research online: a púca is a mischievous fae that often appears in the form of an animal, and occasionally likes to run around at top speed terrorizing others, knocking things over, and generally wreaking havoc. He’s six now and still hasn’t outgrown that. 
We named one of our cats Fancy Meat after an inside joke. Our other cat is Char (short for Charbon).
One of my friends is thinking of getting a second Newfouldland dog, a female, and naming it Aretha Frankin.
We used to spend Easter in Barcelona, at my grandparents’. On the year I was 10, Auntie and Mom took my same-age cousin and myself to a place we’d never been. It was a flat where a woman had many cages with canaries, including a walk-in one. We admired the birds and then Aunt got one for my cousin and one for me!
They asked what was I going to call him, and I asked “since he’s yellow and Catalan… how do you say ‘yellow’ in Catalan?” And thus was Groc named!
My wife and I got Liza, name included, from the shelter. We thought about changing her name, but she already responded to Liza, and she’s really not all that smart, so changing her name would be tricky. So, instead we’ve given her an endless stream of nicknames: Boo, Bubba, Bub, Bean, Marine, Dogface, Dummyhead, Poo and so forth. She doesn’t really seem to mind what we call her as long as she gets scratches on the belly and the occasional pretzel dropped on the floor.
Perfect.
My current critter-naming method is to use a human name, but one with some meaning. Thus we have Bruno (brown), Lloyd (gray), and Lilith (female demon).
Various pets over the last decade or so:
Daisy-Bob. My dog. She lives with my sister now (sniff). I got her in SF when I was married. We’d had a neighbor who had a bulldog named Rosie. So we were fixated on flower names. We were going over a few on the way back from the pound alphabetically. So after Aster, Buttercup and Crysanthymum all failed, we went with Daisy. Bob for a middle name because I always wanted a dog named Bob.
Teddy and The Kaiser. Fish (both deceased). I named them after bad guys in movies I owned. Teddy after Teddy KGB in Rounders and The Kaiser after Keyser Soze in The Usual Suspects.
Morty. My cat. Named after a ficticious 80 year old Jewish relative (and I’m not even Jewish) that my uncle made up in a drunken and hilarious eulogy one Christmas in his best Georgie Jessel impersonation. It’s harder to explain that it is to understand.
Oh. I also once had a hamster named Lunchmeat.
My mother named him. She said, “If that little bastard gets out of his cage one more time, I swear to God he’s lunchmeat.”
With the outdoor cats, it was just for fun. We started with Pinky, because, well, he’s pink. From there, his siblings became Inky, Blinky and Clyde (funny enough, Clyde was the only female in that litter). Then came Garfield (who looks like his namesake) and to keep the names sensible – Odie and Nermal. The last batch were Eeny, Meeny, Miny and Moo.
For the dogs – who are actually pets, as opposed to the cats that are just encouraged to stick around to eat vermin and keep the snake population low – we let our (then 3 year old) son name the Chihuahua – Chili Dog. With the wolf, we actually gave it some thought. We decided to go with the Cherokee for “little brother” since he is the newest member of our pack, and our little brother – Usdi Udo. we call him Udo for short.
In the past, we generally let the cats name themselves. We have Momma Cat – who is the first stray that adopted us when she was pregnant – and Evil Kitty – who ran all of her siblings off so she would be the only one still nursing – and Mr. Moo Cow – who is a big old fatty with cow markings that surprised us by giving birth when he was 2 years old. Yes, we know he is a “she” but he will always be a he to us, lol!
Our one-year-old labradusky is named “Abby”. It’s from “Young Frankenstein”
Our first cat, Muffin, was named because when we first got her, we’d carry her around and she would drape herself over our hands like a muff. Only after she got her name did she devour a pan of blueberry muffins
Max was very distinguished looking, with his white bib and tucker markings, so he became Maximillian, and Cleo was petite and black and slinky, so she was Cleopatra. When Max ran off, Chloe moved in…I liked the juxtaposition of Cleo and Chloe…my family has never forgiven me for it.
Moose…well, Muffin’s original name was Moose when we got her, but that was so wrong for such a dainty little thing. Then a stray adopted Muffin, becoming her devoted slave. He was a big bruiser with torn ears, a 13" neck (we bought dog collars for him) and the sweetest disposition…Moose was the perfect name for him…the big, dopey football player that follows the cute girl around.
We didn’t know much about our current cats before we got them. We got them through my sister’s classmate’s neighbour.
The female one was white, and since we’re somewhat fans of Alice, she got preemptively named Dinah. We also took in her brother, as they were the last ones of that litter yet to find a home. Him, to stay on the literary track, we named Tigger, after Winnie the Pooh.
Two named cats, and no-one but my sister had seen them at that point. All right, one of my parents had as well by then, but still.
My parents named our childhood dog, Lucy. They’d been saving that name for a future daughter but never had one, so they gave up and let the dog have it. And then my mom got pregnant only weeks later, which could have made for an awkward situation, but she had another son.
Current dog #1 was named Maggie for no particular reason. My mom was tired of giving people names to the pets, but nobody liked any of her other ideas (Scout was the best) and she suggested that name out of frustration.
Dog #2 was named Brodie, after a character from Mallrats. I’d moved out by this point and thought it was a terrible choice - it’s a bad movie and the name doesn’t suit the dog, but no dice. He’s an adoptee, and his original name was Desi. Same breed as Lucy, but my family didn’t care for the coincidence.
Dog #3/4 is named Jack just because everybody liked the name. It fits.
My youngest beagle is the sweetest dog I ever saw, but she kept getting in trouble . She tried to do right but things went wrong for her. i named her Nordberg after OJs character in Naked Guns. He would show up when they were supposed to and the rest would be late. He would do the right thing and get in a fix.
I had a siamese cat who was very gentle. My previous siamese were aggressive fighters. One day I I was looking for something ,and I thought .I have looked through 14 drawers and haven’t found my watch. It was just a number that sounded like a lot but really wasn’t quite true. The cat became Fourteen. He was a siamese but not quite. He was a lot of nothing. He was great cat and lived 20 years.
Similar to my parents’ friend, whose cat was named Cholent because the friend’s roomate kept threatning to put the at in the stewpot.
Ariel was named by someone else, and I decided to keep it. She’s obsessed with all things water… she drinks like a fish, comes running when you flush the toilet to watch the water swirl in the bowl, and used to sleep in the bathroom sink. So I guess The Little Mermaid’s name seemed like a good choice.
Morgan named herself. She was originally named “Comet” by the shelter, which didn’t suit her at all, so I was just rattling off possible alternate names to The Boy to see what he’d like… she merped when I said “Morgan?” and it’s stuck ever since. (I’d expect no less from her, since she’s an opinionated little thing… we brought her home from the shelter in the first place because she reached out of her cage and poked me when I got up to look at other cats)
I like the idea of a cat named Cholent. That’s cute. 
I’ve named 4 of 5 out of my current crop of critters. Mollyjust looked like a Molly, so I named her before she was ready to come live with me. Her littermate was named Pumpkin, which got shortened to Punkyby me. I hadn’t planned on taking more than one kitten from the litter, so the people who had the litter named him.
Rio’sname came first, before the cat. I thought that a kitty named Rio, who danced upon the sand, was a great idea.
Jasminecame to live with us when the confederate jasmine was blooming (and killing my allergies.) I wanted a name for my girl that was feminine, but not TOO feminine. It gets shortened to Jazzy a lot.
Sunshineis yellow and chirpy and happy. (And an absolute fiend for mashed potatoes–not that that has anything to do with her name.)
My first cocker spaniel (now deceased) was named Lady because I was reading a biography of Lady Jane Grey when I got her. This led to 14 years of “Lady and the Tramp” references but whatever.
Linus (now four) was named Linus because I was looking for a good male name that started with “L” to go with Lady. But then Lady died 2 weeks after Linus arrived.
Lucy (also now four) was named Lucy to go with Linus. Both are blond American cocker spaniels.
My home bettas are named for the saint whose day it is whatever day I get them. The current one is named Greg, the one before that was named Hilary.
My work betta is named Estelle V. Gamble, which is a dumb inside lawyer’s joke.